1. ## Abstract Algebra- Pre-Exam

Well , I'm trying to solve some exams, and I have some difficulties in the following questions:

1. Let A = where p,q are disjoint primes.
Prove that there are unique m,n such as n|m and A is isomorphic to:
.

2. Let F be a field.
Prove that $\displaystyle \frac{F[x_{1},x_{2}]}{(x_{1})} isomorphic F[x_{2}]$

I realy need your guidance in these ones.

2. Originally Posted by WannaBe
Well , I'm trying to solve some exams, and I have some difficulties in the following questions:

1. Let A = where p,q are disjoint primes.
Prove that there are unique m,n such as n|m and A is isomorphic to:
.

The answer is $\displaystyle n=p, m=p^2q$. Search google with "torsion coefficients" or "invariant factors" for finitely generated abelian groups. I'll give another example. For $\displaystyle \mathbb{Z}_6 \times \mathbb{Z}_{12} \times Z_{20} \cong \mathbb{Z}_{2} \times \mathbb{Z}_{3} \times \mathbb{Z}_{3}\times \mathbb{Z}_{4} \times \mathbb{Z}_{4} \times \mathbb{Z}_{5}$, you can decompose like,

$\displaystyle \begin{matrix} 2&4&4\\ &3&3\\ &&5 \end{matrix}$.

Torsion coefficients (multiply numbers in each column) are 2, 12, 60 where 2|12 and 12|60.

2. Let F be a field.
Prove that $\displaystyle \frac{F[x_{1},x_{2}]}{(x_{1})} isomorphic F[x_{2}]$

I realy need your guidance in these ones.

Let $\displaystyle f:F[x_1, x_2] \rightarrow F[x_2]$, where $\displaystyle x_1 \mapsto 0, x_2 \mapsto x_2$. You can check that f is a surjective group homomorphism. Now the kernel of f is $\displaystyle (x_1)$, the result follows from the first isomorphism theorem.

3. Thanks! Just to make sure:
In $\displaystyle F[x_{1},x_{2}]$, every element has the form:
$\displaystyle f(x)*x_{1}+g(x)*x_{2}$ , right?

Thanks again!

4. Originally Posted by WannaBe
Thanks! Just to make sure:
In $\displaystyle F[x_{1},x_{2}]$, every element has the form:
$\displaystyle f(x)*x_{1}+g(x)*x_{2}$ , right?

Thanks again!
$\displaystyle F[x_{1},x_{2}]$ is the ring of polynomials in two variables x_1, x_2 over F. x is not a variable in this ring.

5. Yep, you're right...So what's the form of the poly. in this ring?

Thanks!

6. Originally Posted by WannaBe
Yep, you're right...So what's the form of the poly. in this ring?

Thanks!
$\displaystyle F[x_1, x_2]=F[x_1][x_2]=\{\sum_{i,j>=0}a_{i,j}x_1^ix_2^j |a_{ij} \in F\}$, where $\displaystyle a_{ij}=0$ all but finite number of values of i and j.

7. ## Subring without identity?

oops, sorry, posted something in the wrong place.

8. thanks a lot!